Let a function f: N N be defined by f(n) = n + n , where x denotes the greatest integer less than or equal to x . Then, f is

Step-by-step solution

For injectivity (one-one): Let n₁, n₂ N with n₁ Since n₁ Adding these inequalities, n₁ + n₁ Since f(n) is strictly increasing, it is a one-one function. For surjectivity (onto): Let us evaluate the first few terms of the function: f(1) = 1 + 1 = 2 f(2) = 2 + 1.414 = 3 f(3) = 3 + 1.732 = 4 f(4) = 4 + 2 = 6 Notice that there is no n N for which f(n) = 1 or f(n) = 5 . Since the range of f does not contain all natural numbers (e.g., 1, 5 Range ), f is not onto. Therefore, f is one-one but not onto. Answer: One-one but